Intution behind the gradient giving the steepest ascent in 2D

In summary, the gradient of a function is a vector that gives the rate of change of the function in a given direction. It is perpendicular to the level curves of the function and the direction of steepest ascent. This can be visualized by imagining walking along a level path on a hill, where the steepest slope is perpendicular to the path. The concept of steepest descent and conjugate gradient methods can be further understood through resources like the article from Better Explained and videos from Khan Academy.
  • #1
Mohankpvk
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Can this(image) be used as a proof that the direction of gradient gives the direction of steepest ascent(in 2D).Am I understanding it right ?.The function 'f' in the image is a scalar valued function.Please explain.
 

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the gradient is a vector whose dot product with a given direction vector gives the rate of change of the function in that direction. hence it dots to zero along a direction where the function is constant, and hence the gradient is perpendicular to the level curves (for a function defined on the plane) of the function. It seems obvious that the direction in which the function increases fastest is perpendicular to the level curve through a given point. just imagine you are walking along a level path cut into the side of a hill, wouldn't the slope of the hill be greatest in a direction perpendicular to the path?
 
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1. What is the gradient in 2D and how does it relate to intuition?

The gradient in 2D is a mathematical concept that represents the rate of change of a function in two dimensions. It is a vector that points in the direction of the steepest ascent of the function. Intuitively, this means that if you were to walk in the direction of the gradient, you would be moving in the direction that leads to the highest values of the function.

2. Why is the gradient important in optimization problems?

The gradient is important in optimization problems because it allows us to find the maximum or minimum value of a function. By following the direction of the gradient, we can find the steepest ascent or descent of the function, which leads us to the optimal solution.

3. How does the gradient give the steepest ascent in 2D?

The gradient gives the steepest ascent in 2D by pointing in the direction of the greatest increase of the function. This means that if you were to take a step in the direction of the gradient, you would be moving in the direction that leads to the highest values of the function.

4. Can the gradient give the steepest ascent in 2D for any type of function?

Yes, the gradient can give the steepest ascent in 2D for any differentiable function. This means that as long as the function has a continuous slope, the gradient will accurately point in the direction of the steepest ascent.

5. How does the gradient relate to the concept of slope?

The gradient is similar to the concept of slope, but it is multidimensional. In 2D, the gradient can be thought of as the slope of the function in the x and y directions. However, in higher dimensions, the gradient takes into account the rate of change in all directions, making it a more comprehensive measure of the function's behavior.

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